A new result on Chvátal's conjecture
type: Item
instance of: scholarly article
title: A new result on Chvátal's conjecture
zbMATH Open document ID: 0760.05087
DOI: 10.1016/0097-3165(92)90059-4
published in: Journal of Combinatorial Theory. Series A
publication date: 1992-10-05T00:00:00Z
review text: Let \(I\) be an ideal of the finite set \(X\). An element \(j\in X\) is a dominant element of \(I\), if \(j\notin A\in I\) implies \(A\backslash\{a\}\cup\{j\}\in I\) for every \(a\in A\). (This is a generalization of ideals equiped by a hereditary tree structure, introduced by \textit{D.-L. Wang} and \textit{P. Wang} [Discrete Math. 24, 95- 101 (1978; Zbl 0384.05003)].) The paper's main result shows that Chvátal's conjecture [Proc. 1st Working Sem. Hypergraphs, Columbus 1972, Lect. Notes Math. 411, 61-66 (1974; Zbl 0305.05128)] holds for ideals having dominant elements. The paper also proves that if the bases of the ideal \(I\) can be partitioned into two strong delta systems with one-element kernels, then the ideal \(I\) satisfies Chvátal's conjecture.
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 68491
zbMATH Keywords: Chvátal's conjecture
zbMATH Keywords: ideal
zbMATH Keywords: intersecting set systems
zbMATH Keywords: shifting technique
zbMATH Keywords: star
Wikidata QID: Q123151142
author: Hunter Snevily
reviewed by: Péter L. Erdős
MaRDI profile type: Publication
full work available at URL: 0097-3165(92)90059-4
OpenAlex ID: W1986108838
cites work: Q3741626
cites work: Q4060998
cites work: Combinatorial Optimization Problems for Systems of Subsets
cites work: On the number of latent subsets of intersecting collections
cites work: On Chvatal's conjecture related to a hereditary system
cites work: Some results about the Chvatal conjecture
cites work: Q3964569
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description: scientific article; zbMATH DE number 68491